Question: Simplify the following expression: $r = \dfrac{120t^3 + 36t^2}{24t^3 + 108t^2}$ You can assume $t \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $120t^3 + 36t^2 = (2\cdot2\cdot2\cdot3\cdot5 \cdot t \cdot t \cdot t) + (2\cdot2\cdot3\cdot3 \cdot t \cdot t)$ The denominator can be factored: $24t^3 + 108t^2 = (2\cdot2\cdot2\cdot3 \cdot t \cdot t \cdot t) + (2\cdot2\cdot3\cdot3\cdot3 \cdot t \cdot t)$ The greatest common factor of all the terms is $12t^2$ Factoring out $12t^2$ gives us: $r = \dfrac{(12t^2)(10t + 3)}{(12t^2)(2t + 9)}$ Dividing both the numerator and denominator by $12t^2$ gives: $r = \dfrac{10t + 3}{2t + 9}$